Quotient structure ($A/\sim$)

For an algebraic structure $A$ specified by operations, a quotient structure identifies elements through an equivalence relation $\sim$ compatible with every operation. Explicitly, if $a_i\sim b_i$ for all inputs, compatibility requires $f(a_1,\dots ,a_n)\sim f(b_1,\dots ,b_n)$. Thenis independent of the chosen representatives. The map $a\mapsto [a]$ is a surjective homomorphism.

For the integers, congruence modulo $m$ is compatible with addition and multiplication, giving the quotient ring $\mathbb{Z}/m\mathbb{Z}$. For a group, equivalence by cosets supports a quotient group precisely when the subgroup is normal. The compatibility requirement is what makes the quotient a reusable algebraic object rather than merely a partition of a set.